Let \(M\) be a closed irreducible 3-manifold and \(f\in \text{Diff} (M)\) a gradient-like diffeomorphism on \(M\). By \(\Omega (f)\) we denote the set of periodic points of \(f\). Represent the set of saddle periodic points as the union \(\Omega_1(f) \cup \Omega_2 (f)\), where \(\Omega_1(f)\) \((\Omega_2(f))\) consists of all points \(p\) such that \(\dim W^u(p)=1\) \((\dim W^u(p) =2)\). The graph \(G(f)\) of a diffeomorphism \(f\) is an oriented graph whose vertices correspond to points from \(\Omega(f)\) and whose edges correspond to connected components of the set \[ (W^s \cup W^u) \backslash \bigl(\Omega_1 (f)\cup \Omega_2(f) \bigr),\quad \text{where} \quad W^\sigma= \cup W^\sigma (p),\;\sigma \in\{s,u\}. \] A subgraph of the graph \(G (f)\) is assigned to the boundary of a connected component \(K\) of the set \(M \backslash K\). This subgraph is called the distinguishing set of the graph \(G(f)\). The distinguishing graph \(G^* (f)\) of the diffeomorphism \(f\) is a graph \(G(f)\) with the system of marked subgraphs that are distinguishing sets. Further, the isomorphism between the graphs \(G^*(f)\) and \(G^*(g)\) is described. It is easy to see, that the diffeomorphism \(f\) induces the permutation \(P(f)\) on the set of the vertices of the graph \(G(f)\). The authors prove that gradient-like diffeomorphisms \(f\) and \(g\) are topologically conjugate iff there exists an isomorphism \(\varphi\) of the graphs \(G^*(f)\) and \(G^*(g)\), such that \(P(g)= \varphi P(f) \varphi^{-1}\).